TY - JOUR
AB - We consider the totally asymmetric simple exclusion process in a critical scaling parametrized by a≥0, which creates a shock in the particle density of order aT−1/3, T the observation time. When starting from step initial data, we provide bounds on the limiting law which in particular imply that in the double limit lima→∞limT→∞ one recovers the product limit law and the degeneration of the correlation length observed at shocks of order 1. This result is shown to apply to a general last-passage percolation model. We also obtain bounds on the two-point functions of several airy processes.
AU - Nejjar, Peter
ID - 70
IS - 2
JF - Latin American Journal of Probability and Mathematical Statistics
SN - 1980-0436
TI - Transition to shocks in TASEP and decoupling of last passage times
VL - 15
ER -
TY - JOUR
AB - We consider products of independent square non-Hermitian random matrices. More precisely, let X1,…, Xn be independent N × N random matrices with independent entries (real or complex with independent real and imaginary parts) with zero mean and variance 1/N. Soshnikov-O’Rourke [19] and Götze-Tikhomirov [15] showed that the empirical spectral distribution of the product of n random matrices with iid entries converges to (equation found). We prove that if the entries of the matrices X1,…, Xn are independent (but not necessarily identically distributed) and satisfy uniform subexponential decay condition, then in the bulk the convergence of the ESD of X1,…, Xn to (0.1) holds up to the scale N–1/2+ε.
AU - Nemish, Yuriy
ID - 1023
JF - Electronic Journal of Probability
SN - 10836489
TI - Local law for the product of independent non-Hermitian random matrices with independent entries
VL - 22
ER -
TY - JOUR
AB - The eigenvalue distribution of the sum of two large Hermitian matrices, when one of them is conjugated by a Haar distributed unitary matrix, is asymptotically given by the free convolution of their spectral distributions. We prove that this convergence also holds locally in the bulk of the spectrum, down to the optimal scales larger than the eigenvalue spacing. The corresponding eigenvectors are fully delocalized. Similar results hold for the sum of two real symmetric matrices, when one is conjugated by Haar orthogonal matrix.
AU - Bao, Zhigang
AU - Erdös, László
AU - Schnelli, Kevin
ID - 1207
IS - 3
JF - Communications in Mathematical Physics
SN - 00103616
TI - Local law of addition of random matrices on optimal scale
VL - 349
ER -
TY - JOUR
AB - We consider the local eigenvalue distribution of large self-adjoint N×N random matrices H=H∗ with centered independent entries. In contrast to previous works the matrix of variances sij=\mathbbmE|hij|2 is not assumed to be stochastic. Hence the density of states is not the Wigner semicircle law. Its possible shapes are described in the companion paper (Ajanki et al. in Quadratic Vector Equations on the Complex Upper Half Plane. arXiv:1506.05095). We show that as N grows, the resolvent, G(z)=(H−z)−1, converges to a diagonal matrix, diag(m(z)), where m(z)=(m1(z),…,mN(z)) solves the vector equation −1/mi(z)=z+∑jsijmj(z) that has been analyzed in Ajanki et al. (Quadratic Vector Equations on the Complex Upper Half Plane. arXiv:1506.05095). We prove a local law down to the smallest spectral resolution scale, and bulk universality for both real symmetric and complex hermitian symmetry classes.
AU - Ajanki, Oskari H
AU - Erdös, László
AU - Krüger, Torben H
ID - 1337
IS - 3-4
JF - Probability Theory and Related Fields
SN - 01788051
TI - Universality for general Wigner-type matrices
VL - 169
ER -
TY - JOUR
AB - We prove a local law in the bulk of the spectrum for random Gram matrices XX∗, a generalization of sample covariance matrices, where X is a large matrix with independent, centered entries with arbitrary variances. The limiting eigenvalue density that generalizes the Marchenko-Pastur law is determined by solving a system of nonlinear equations. Our entrywise and averaged local laws are on the optimal scale with the optimal error bounds. They hold both in the square case (hard edge) and in the properly rectangular case (soft edge). In the latter case we also establish a macroscopic gap away from zero in the spectrum of XX∗.
AU - Alt, Johannes
AU - Erdös, László
AU - Krüger, Torben H
ID - 1010
JF - Electronic Journal of Probability
SN - 10836489
TI - Local law for random Gram matrices
VL - 22
ER -
TY - JOUR
AB - We consider N×N Hermitian random matrices H consisting of blocks of size M≥N6/7. The matrix elements are i.i.d. within the blocks, close to a Gaussian in the four moment matching sense, but their distribution varies from block to block to form a block-band structure, with an essential band width M. We show that the entries of the Green’s function G(z)=(H−z)−1 satisfy the local semicircle law with spectral parameter z=E+iη down to the real axis for any η≫N−1, using a combination of the supersymmetry method inspired by Shcherbina (J Stat Phys 155(3): 466–499, 2014) and the Green’s function comparison strategy. Previous estimates were valid only for η≫M−1. The new estimate also implies that the eigenvectors in the middle of the spectrum are fully delocalized.
AU - Bao, Zhigang
AU - Erdös, László
ID - 1528
IS - 3-4
JF - Probability Theory and Related Fields
SN - 01788051
TI - Delocalization for a class of random block band matrices
VL - 167
ER -
TY - JOUR
AB - We consider last passage percolation (LPP) models with exponentially distributed random variables, which are linked to the totally asymmetric simple exclusion process (TASEP). The competition interface for LPP was introduced and studied in Ferrari and Pimentel (2005a) for cases where the corresponding exclusion process had a rarefaction fan. Here we consider situations with a shock and determine the law of the fluctuations of the competition interface around its deter- ministic law of large number position. We also study the multipoint distribution of the LPP around the shock, extending our one-point result of Ferrari and Nejjar (2015).
AU - Ferrari, Patrik
AU - Nejjar, Peter
ID - 447
JF - Revista Latino-Americana de Probabilidade e Estatística
TI - Fluctuations of the competition interface in presence of shocks
VL - 9
ER -
TY - JOUR
AB - We prove the universality for the eigenvalue gap statistics in the bulk of the spectrum for band matrices, in the regime where the band width is comparable with the dimension of the matrix, W ~ N. All previous results concerning universality of non-Gaussian random matrices are for mean-field models. By relying on a new mean-field reduction technique, we deduce universality from quantum unique ergodicity for band matrices.
AU - Bourgade, Paul
AU - Erdös, László
AU - Yau, Horng
AU - Yin, Jun
ID - 483
IS - 3
JF - Advances in Theoretical and Mathematical Physics
SN - 10950761
TI - Universality for a class of random band matrices
VL - 21
ER -
TY - JOUR
AB - For large random matrices X with independent, centered entries but not necessarily identical variances, the eigenvalue density of XX* is well-approximated by a deterministic measure on ℝ. We show that the density of this measure has only square and cubic-root singularities away from zero. We also extend the bulk local law in [5] to the vicinity of these singularities.
AU - Alt, Johannes
ID - 550
JF - Electronic Communications in Probability
SN - 1083589X
TI - Singularities of the density of states of random Gram matrices
VL - 22
ER -
TY - BOOK
AB - This book is a concise and self-contained introduction of recent techniques to prove local spectral universality for large random matrices. Random matrix theory is a fast expanding research area, and this book mainly focuses on the methods that the authors participated in developing over the past few years. Many other interesting topics are not included, and neither are several new developments within the framework of these methods. The authors have chosen instead to present key concepts that they believe are the core of these methods and should be relevant for future applications. They keep technicalities to a minimum to make the book accessible to graduate students. With this in mind, they include in this book the basic notions and tools for high-dimensional analysis, such as large deviation, entropy, Dirichlet form, and the logarithmic Sobolev inequality.
AU - Erdös, László
AU - Yau, Horng
ID - 567
SN - 9781470436483
TI - A dynamical approach to random matrix theory
VL - 28
ER -